BetterGrades Precalculus · Unit 6 · Lesson

Rational inequalities

Solve rational inequalities using critical values, sign intervals, and correct endpoint inclusion.

Opening

Start with the situation

Rational inequalities are solved by critical-value sign intervals rather than unsafe cross multiplication.

Rational graphs are organized around the inputs the denominator forbids. Those exclusions divide the graph into continuity intervals and explain why a simplified expression may still contain a hole or asymptote.

Before you begin

Prerequisite check

Factor numerator and denominator.
Preserve original restrictions.
Use sign and asymptotic notation.

Core explanation

Explanation

Move all terms to one side, combine and factor, list numerator and denominator zeros, test intervals, and apply endpoint rules.

Allowed numerator zeros may be included for non-strict inequalities; denominator zeros never are.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through inequality sign chart, endpoint rules, or another equivalent representation.

Conceptual reading

What the idea is really doing

A rational function carries permanent memory of its original denominator. Factoring may reveal holes, asymptotes, and sign changes, but cancellation never restores an input that the original formula excluded.

This lesson narrows that lens to one goal: solve rational inequalities using critical values, sign intervals, and correct endpoint inclusion. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Move all terms to one side.
Combine.
Factor.
List numerator.

Verification: Record exclusions first, then compare the factored and simplified forms. Test one point in every sign interval and examine both sides of each vertical asymptote.

Foundation walkthrough

Plan before calculating

Problem

Solve $\frac{x+3}{x-4}\le 0$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Move all terms to one side, combine and factor, list numerator and denominator zeros, test intervals, and apply endpoint rules.
Conclusion: $[-3,4)$
Why the check works: Include the zero, exclude the asymptote.

Worked examples

See the idea in three forms

foundation example

Solve $\frac{x+3}{x-4}\le 0$

Solution $[-3,4)$

Include the zero, exclude the asymptote.

representation example

Solve $\frac{1}{x}<0$

Solution $(-∞,0)$

This example expresses rational inequalities in a second form.

transfer example

Which endpoints may be included?

SolutionAllowed numerator zeros.

Allowed numerator zeros may be included for non-strict inequalities; denominator zeros never are.

Inequality sign chart. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Include the zero, exclude the asymptote. — **Anchor figure · Inequality sign chart**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Include the zero, exclude the asymptote.

Endpoint rules. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational inequalities. — **Mechanism figure · Endpoint rules**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational inequalities.

Function comparison. Compare the valid path with the tempting shortcut. The figure shows why multiplying by a denominator whose sign changes across the domain leads to a false conclusion. — **Comparison and error figure · Function comparison**
Compare the valid path with the tempting shortcut. The figure shows why multiplying by a denominator whose sign changes across the domain leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is multiplying by a denominator whose sign changes across the domain.

Check yourself

Solve $\frac{x-2}{x+5}<0$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Solve $\frac{x-2}{x+5}<0$

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Solve $\frac{1}{x}<0$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Which endpoints may be included?

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Practice 404

Which always excluded?

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Practice 505

Explain why this conclusion is valid: $[-3,4)$ . Use the foundation problem as evidence: Solve $\frac{x+3}{x-4}\le 0$ .

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Practice 606

Solve the representation example, then name the feature of rational inequalities that it illustrates: Solve $\frac{1}{x}<0$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is multiplying by a denominator whose sign changes across the domain.

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Practice 808

Connect two representations for this example: Solve $\frac{x+3}{x-4}\le 0$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Which endpoints may be included? Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for rational inequalities, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Variation and rational models, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Solvex+3x−4≤0\frac{x+3}{x-4}\le 0x−4x+3​≤0

Solvex+3x−4≤0\frac{x+3}{x-4}\le 0x−4x+3​≤0

Solve1x<0\frac{1}{x}<0x1​<0

Which endpoints may be included?

Original BetterGrades manuscript, rights-separated references.

Solve $\frac{x+3}{x-4}\le 0$

Solve $\frac{x+3}{x-4}\le 0$

Solve $\frac{1}{x}<0$